Counterdiabaticity and the quantum approximate optimization algorithm

Jonathan Wurtz; Peter J. Love

doi:10.22331/q-2022-01-27-635

Counterdiabaticity and the quantum approximate optimization algorithm

Jonathan Wurtz and Peter J. Love

Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA

Published:	2022-01-27, volume 6, page 635
Eprint:	arXiv:2106.15645v3
Doi:	https://doi.org/10.22331/q-2022-01-27-635
Citation:	Quantum 6, 635 (2022).

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Abstract

The quantum approximate optimization algorithm (QAOA) is a near-term hybrid algorithm intended to solve combinatorial optimization problems, such as MaxCut. QAOA can be made to mimic an adiabatic schedule, and in the $p\to\infty$ limit the final state is an exact maximal eigenstate in accordance with the adiabatic theorem. In this work, the connection between QAOA and adiabaticity is made explicit by inspecting the regime of $p$ large but finite. By connecting QAOA to counterdiabatic (CD) evolution, we construct CD-QAOA angles which mimic a counterdiabatic schedule by matching Trotter "error" terms to approximate adiabatic gauge potentials which suppress diabatic excitations arising from finite ramp speed. In our construction, these "error" terms are helpful, not detrimental, to QAOA. Using this matching to link QAOA with quantum adiabatic algorithms (QAA), we show that the approximation ratio converges to one at least as $1-C(p)\sim 1/p^{\mu}$. We show that transfer of parameters between graphs, and interpolating angles for $p+1$ given $p$ are both natural byproducts of CD-QAOA matching. Optimization of CD-QAOA angles is equivalent to optimizing a continuous adiabatic schedule. Finally, we show that, using a property of variational adiabatic gauge potentials, QAOA is at least counterdiabatic, not just adiabatic, and has better performance than finite time adiabatic evolution. We demonstrate the method on three examples: a 2 level system, an Ising chain, and the MaxCut problem.

Featured image: Schematic diagram of the CD-QAOA angle matching procedure. As input, the procedure receives the counterdiabatic annealing protocol (Black line), effective Hamiltonian H, and number of QAOA steps p. For each step of the CD-QAOA evolution (pink), the effective unitaries over the interval are matched by minimizing the difference of their generators (right). This procedure sets the value of each angle γ, β in the QAOA (blue). By best matching along each step, the procedure fixes the angles {γ, β} and total adiabatic annealing time T.

► BibTeX data

@article{Wurtz2022counterdiabaticity,
  doi = {10.22331/q-2022-01-27-635},
  url = {https://doi.org/10.22331/q-2022-01-27-635},
  title = {Counterdiabaticity and the quantum approximate optimization algorithm},
  author = {Wurtz, Jonathan and Love, Peter J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {635},
  month = jan,
  year = {2022}
}

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